Continuum mechanics principles are applied to standard balance equations assuming constitutive equations carried out to first derivatives of density, temperature, and velocity. Using the principles of isotropy, frame invariance, and equipresence, we obtain constitutive equations for the Helmholtz free energy, entropy, entropy flux, extra entropy flux, and Cauchy stress tensor that satisfy the Clausius-Duhem inequality. The equation for the Cauchy stress tensor contains a term that is missing in the well-known Navier-Stokes-Fourier equations. The additional term, when applied to an ideal gas, appears to be due to the local time relaxation of molecular vibrational energy and leads to a new definition of the bulk viscosity. We also obtain additional terms in the entropy that are due to the relative time rate of change of the local volume.

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• Compressible flows
• Laminar flows
• Rarefied flows
• Navier-Stokes equation